arXiv subjects
Publications and source records attributed to Julia E. Bergner.
We show that various categories of trees can be modeled by Grothendieck constructions on categories of trees with a fixed set of leaves. We prove this result for the dendroidal category $Ω$, the category $Ω^G$ of trees with a $G$-action for a finite group $G$, and finally for the category of genuine equivariant trees $Ω_G$ that has played an important role in recent work on genuine equivariant operads.
In this paper, we establish a precise relationship between CGW categories and pointed stable double Segal spaces, both of which were developed as general input for algebraic K-theory. In particular, we show that any CGW category can be regarded as a pointed stable double Segal space, and they can be identified using a classifying diagram construction for double categories with shared isomorphisms.
We introduce two definitions of $G$-equivariant partitions of a finite $G$-set, both of which yield $G$-equivariant partition complexes. By considering suitable notions of equivariant trees, we show that $G$-equivariant partitions and $G$-trees are $G$-homotopy equivalent, generalizing existing results for the non-equivariant setting. Along the way, we develop equivariant versions of Quillen's Theorems A and B, which are of independent interest.
We establish cartesian model structures for variants of $Θ_n$-spaces in which we replace some or all of the completeness conditions by discreteness conditions. We prove that they are all equivalent to each other and to the $Θ_n$-space model, and we give a criterion for which combinations of discreteness and completeness give non-overlapping models. These models can be thought of as generalizations of Segal categories in the framework of $Θ_n$-diagrams. In the process, we give a characterization of the Dwyer-Kan equivalences in the $Θ_n$-space model, generalizing the one given by Rezk for complete Segal spaces.
The theory of 2-Segal sets has connections to various important constructions such as the Waldhausen $S_\bullet$-construction in algebraic $K$-theory, Hall algebras, and (co)operads. In this paper, we construct 2-Segal sets from rooted trees and explore how these applications are illustrated by this example.
We give an introduction to the theory of 2-Segal sets, and two of the main applications of them: Hall algebras and a discrete version of Waldhausen's $S_\bullet$-construction. We present several combinatorial examples and how these constructions can be applied to them.
The definition of the homotopy limit of a diagram of left Quillen functors of model categories has been useful in a number of applications. In this paper we review its definition and summarize some of these applications. We conclude with a discussion of why we could work with right Quillen functors instead, but cannot work with a combination of the two.
The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of higher categorical methods in a wide range of applications, it is important for researchers in a range of fields to have a good working knowledge of them. This paper is intended as an introduction to simplicial sets, both as an overview of their development from other concepts, and as a user's guide for someone wanting to read modern literature that makes use of them.
Model structures for many different kinds of functor calculus can be obtained by applying a theorem of Bousfield to a suitable category of functors. In this paper, we give a general criterion for when model categories obtained via this approach are cofibrantly generated. Our examples recover the homotopy functor and $n$-excisive model structures of Biedermann and Röndigs, with different proofs, but also include a model structure for the discrete functor calculus of Bauer, Johnson, and McCarthy.
In this survey article, we review some conceptual approaches to the cyclic category $Λ$, as well as its description as a crossed simplicial group. We then give a new proof of the model structure on cyclic sets, work through the details of the generalized Reedy structure on cyclic spaces, and introduce model structures for cyclic Segal spaces and cyclic 2-Segal spaces.
In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.
We produce a direct Quillen equivalence between two models of $(\infty,2)$-categories: the complete Segal $Θ_2$-spaces due to Rezk and the $2$-complicial sets due to Verity.
In previous work, we develop a generalized Waldhausen $S_{\bullet}$-construction whose input is an augmented stable double Segal space and whose output is a unital 2-Segal space. Here, we prove that this construction recovers the previously known $S_{\bullet}$-constructions for exact categories and for stable and exact $(\infty,1)$-categories, as well as the relative $S_{\bullet}$-construction for exact functors.
We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step $k$ is equal to the $k$th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step $k$ and the set of planar rooted trees with $k$ edges.
We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of trees.
We give describe several models for $(\infty,n)$-categories, with an emphasis on models given by diagrams of sets and simplicial sets. We look most closely at the cases when $n \leq 2$, then summarize methods of generalizing for all $n$.
In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_\bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_\bullet$-constructions.
We show that the edgewise subdivision of a $2$-Segal object is always a Segal object, and furthermore that this property characterizes $2$-Segal objects.